Questions and Answers of Accounting For Financial Instruments

Exercise 8.11 Compute the Rosenblatt transform for the bivariate Gumbel copula. Is it invertible explicitly?
Exercise 8.10 Suppose that for a sample of size n, we have ρ(S) = 0.6 ± .03 at the 95% level of confidence. If the underlying copula is the Plackett, find a 95%confidence interval for θ.
Exercise 8.9 Explain how you could generate observations from a Plackett copula for a given τ.
Exercise 8.8 Suppose that for a sample of size n, we have τ = −0.3 ± .04 at the 95%level of confidence. If the underlying copula is the Frank, find a 95% confidence interval for θ.
Exercise 8.7 For each of the following cases, identify the (bivariate) copula family and compute the associated Kendall tau. (i) Co(u,z)= {max (0, + 1)}. P[In(u1) (ii) Co(u1, u2) = exp |In (1) +
Exercise 8.6 Suppose that for a sample of size n, we have τ = 0.3±.08 at the 95% level of confidence. If the underlying copula is the Clayton, find a 95% confidence interval for θ.
Exercise 8.5 For θ > 0, we set(a) Identify the copula family.(b) Find the following limits:(i) limθ→0 Cθ(u1, u2);(ii) limθ→∞ Cθ(u1, u2);(iii) limθ→−∞ Cθ(u1, u2).(c) For θ
Exercise 8.4 Suppose that τ = 0.5. Find the associated parameter for the following models:(a) Gaussian copula Cρ.(b) Student copula Cρ,ν .(c) Clayton copula Cθ(d) Frank copula Cθ.(e) Gumbel
Exercise 8.3 Suppose that the joint distribution of (X, Y ) is Student with parametersν > 0 and ρ ∈ (−1, 1). Find the distribution of X and the conditional distribution of Y given X = x.
Exercise 8.2 Consider the random variablewhere the Zijs are independent and identically distributed standard Gaussian variables.(a) Find the first six cumulants of X. Note that the famous zeta6
Exercise 8.1 For θ ∈ (−∞,∞), θ = 0, define(a) Identify the copula family.(b) Find the value of the following limits:(i) limθ→0 Cθ(u1, u2);(ii) limθ→∞ Cθ(u1, u2);(iii)
3. Construct a MATLAB function to simulate approximative daily values of the process V satisfying the stochastic differential equation dV (t) = α {β − V (t)} dt + γV (t)dW(t), V(0) = V0 >
2. Construct a MATLAB function to simulate a E-GARCH, a NGARCH, a GARCH-M, and a GJR-GARCH. The innovations can be either Gaussian or GED.
1. Construct a MATLAB function to compute the distribution functionRecall that quantiles of order 90%, 95%, and 99% of F are respectively 1.96, 2.241, and 2.807. F(x)= = k=1 (-1)-(2k+1)/(8), 2k+1 k
Exercise 7.8 Suppose that the estimated parameters of a GARCH(1,1) are μ = 0.0005,ω = 1.5 × 10−7, β = .81 and α = .17. Here we assume the innovations and of the form εi = (i −ν)/√ν,
Exercise 7.7 Consider the random variablewhere Z1, Z2, . . . are independent and identically distributed standard Gaussian variables. Show that the law of X is infinitely divisible. If Y is a L´evy
Exercise 7.6 Suppose that the estimated parameters of a GARCH(1,1) are μ = 0.002, ω = 2.5×10−6, β = .88 and α = .105. Here we assume Gaussian innovations.What are the corresponding parameters
Exercise 7.5 Suppose that the innovations are of the form εi = (i − ν)/√ν, withi ∼ Gamma(ν, 1), with ν > 0. Find the continuous time approximation of a GARCH(1,1) process with these
Exercise 7.4 Suppose that W is a Brownian motion, and that the time scale is in days.Find c so that P sup0≤t≤21|W(t)| ≤ c = 0.95.
Exercise 7.3 For the GARCH(1,1), EGARCH, NGARCH, and GJR-GARCH models, find sufficient conditions for the existence of the limit E σ4 i . Can you find the limit?
Exercise 7.2 Consider the random variable(b) Use the Edgeworth expansion to try to approximate the distribution function of X.(c) Estimate the quantiles of order 90%, 95%, and 99% using the
Exercise 7.1 For the EGARCH, NGARCH, and GJR-GARCH models, find sufficient conditions for the lack of memory property, i.e., for two processes with the same innovations but starting from different
6. Construct a MATLAB function to estimate the parameters of the Normal Inverse Gaussian process model using the cumulants
5. Construct a MATLAB function to estimate the parameters of the Variance Gamma model using the cumulants.
4. Construct a MATLAB function to estimate the value of a European call or put option, under the Variance Gamma model.
3. Construct a MATLAB function for computing the value of a European call or put option for the Merton model.
2. Construct a MATLAB function to compute the new parameters of the Kou model under the change of measure defined by Ub,φ, with φ(x) =(ζ01 + ζ11x)I(x > 0) + (ζ02 + ζ12x)I(x ≤ 0), and b ∈ R.
1. Construct a MATLAB function to compute the new parameters of the Merton model under the change of measure defined by Ub,φ, with φ(x) =ζ0 + ζ1x and b ∈ R. Use the same notations as in Section
Exercise 6.8 Let X be a jump-diffusion process of the Kou type with parameters μ =.08, σ = 0.22, λ = 100, p = 0.7, η1 = 125, and η2 = 100. Suppose that the risk-free rate is 2%, and consider the
Exercise 6.7 Suppose that X is a L´evy process such that X(1) ∼ VG(μ, σ, α). Using its cumulant generating function, show that X(t) ∼ VG(μt, σ√t, αt), t > 0.
Exercise 6.6 Let X be a jump-diffusion process of the Merton type with parametersμ = .08, σ = 0.22, λ = 100, γ = .05 and δ = .005. Suppose that the riskfree rate is 2%, and consider the change
Exercise 6.5 Consider a Normal Inverse Gaussian process with parameters α, β, δ. Using an Esscher transform, what is the law of the process under the new measure?
Exercise 6.4 Suppose X is an Inverse Gaussian process with parameters (α, β) =(20, 0.01). Find its characteristics.
Exercise 6.3 Consider two jump-diffusion Kou models with parameters μ, σ, λ, p, η1,η2 and ˜μ, σ,˜λ, ˜p, ˜η1, ˜η2. Find the change of measure to transform the first one into the second
Exercise 6.2 Suppose X is a Gamma process with parameters (α, β) = (20, 0.01). Find its characteristics.
Exercise 6.1 Consider two jump-diffusion Merton models with parameters μ, σ, λ, γ, δand ˜μ, σ, ˜ λ, ˜γ, δ. Find the change of measure to transform the first one into the second one. Can
1. Find the value of a call option on a zero-coupon bond when the shortterm interest rate is modeled by a Ornstein-Uhlenbeck process. Compute the greeks with respect to α, β, σ, q1, and q2.
Exercise 5.9 In (g) of Exercises 5.6 and 5.8, we assumed a risk-free rate of 2%. However the loss should have been written as X = V0 − eAτ−r(t)∗ Bτ100− 1 100 t0 r(s)ds, with t = 1/12 and τ
Exercise 5.8 Suppose that the short-term rate r (in percentage on a yearly basis) is modeled by a Feller process with parameters α = 0.9, β = 2.5, and σ = 0.95.Suppose also that the market price
Exercise 5.7 What is the effect of increasing the value q2 of the market price of risk in either the CIR or the Vasicek model? Consider only non-negative values of q2.
Exercise 5.6 Suppose that the short-term rate r (in percentage on a yearly basis) is modeled by a Ornstein-Uhlenbeck process with parameters α = 0.5, β = 2 and σ = 0.9. Suppose also that the
Exercise 5.5 Consider the CIR model for the short-term interest rate r.(a) Interpret the so-called mean-reverting property.(b) Show that limt→∞ E{r(t)} = β.(c) Prove that limt→∞ V ar{r(t)} =
Exercise 5.4 Suppose that r is the short-term interest rate (expressed in percentage on a yearly basis) and that it is modeled by a Feller process with parametersα = 0.6, β = 2.25, σ = 0.5.(a)
Exercise 5.3 Consider the Vasicek model for the short-term interest rate r.(a) Interpret the so-called mean-reverting property.(b) Show that limt→∞ E{r(t)} = β.(c) Show that V ar{r(t)} = σ2
Exercise 5.2 Suppose that r is the short-term interest rate (expressed in percentage on a yearly basis) and that it is modeled by a Ornstein-Uhlenbeck process with parameters α = 0.8, β = 3.25, σ
Exercise 5.1 Suppose that ˜r is an Ornstein-Uhlenbeck with parametersa, b, σ under the equivalent martingale measure.(a) Find(b) For any x ∈ R and u ∈ R, findThe last expectation can be useful
2. Construct a MATLAB function to implement the saddlepoint approximation for the random variable Q with cumulant generating function(4.18).
1. Consider a portfolio of m assets composed of European call options with different strikes and maturities, where the underlying assets are correlated geometric Brownian motions. We want to be able
Exercise 4.8 Suppose that over the last 5 years, the monthly returns of a portfolio generated an average return of 0.75% with a 2% volatility. Assuming that the returns are correctly modeled by a
Exercise 4.7(a) Give an interpretation of the following measures of performance:(i) The Sharpe ratio.(ii) The Sortino ratio.(iii) The Omega ratio.(b) Are these measures of performance coherent with
Exercise 4.6 Compute the saddlepoint approximation of the density and the distribution function of a non-central chi-square distribution. Compare the graph of the approximated density with the real
Exercise 4.5 Discuss the advantages and disadvantages of the following approximation methods:(a) The full Monte Carlo methodology.(b) The partial Monte Carlo methodology.(c) The Edgeworth
Exercise 4.4 Assume, as in Exercise 4.2, that the returns of a portfolio loss X at any time t are modeled by a process of the form μ+σW(t), where W is a Brownian motion. Using n = 506 daily
Exercise 4.3(a) Give a financial interpretation for each of the following axioms:(i) Translation invariance : ρ(X +a) = ρ(X) +a, for any a ∈ R.(ii) Monotonicity : X ≤ Y ⇒ ρ(X) ≤ ρ(Y
Exercise 4.2 Assume that the returns of a portfolio loss X at any time t are correctly modeled by a process of the form μ+σW(t), where W is a Brownian motion.Using n = 506 daily returns, the mean
Exercise 4.1(a) Give an interpretation of the following measures of risk:(i) V aR(100p%).(ii) ES(100p%).(b) We want to estimate the VaR of a given portfolio composed of stocks and options on these
2. Construct a MATLAB function for approximating the functions ˇ Ck and ˇak for a call or a put option, when the underlying process is the symmetric Variance Gamma process. This function must
1. Construct a MATLAB function to compute the hedging error of a portfolio with the delta hedging method for replicating a call or a put option, under the Black-Scholes model. The input values
Exercise 3.11 Compute the mean and variance ofwhere the random variables Zi1,...,ik are i.i.d. standard Gaussian. Tk = 2k 11-1 ik=1 Z (ii)'
Exercise 3.10 For the simulation of hedging errors in discrete time for the Black-Scholes model, Wilmott [2006b] uses the following approximation of the standard Gaussian distribution:where U1, . . .
Exercise 3.9 If U1, . . . , Un are i.i.d. uniform variables over (0, 1), find the distribution of n F= -2In(U). i=1
Exercise 3.8 Test the normality hypothesis for the Microsoft data set.
Exercise 3.7 What are the main advantages of optimal hedging over delta hedging? Can you find any negative points against optimal hedging?
Exercise 3.6 For a data set of n = 506 observations, one finds that the estimation of the skewness is −0.15 and the estimation of the kurtosis is 3.4. Test the normality hypothesis using the
Exercise 3.5 Give some reasons why option prices on markets are different from the ones obtained with the Black-Scholes model. For each reason, indicate (a) which assumption is violated in the
Exercise 3.4 Test the serial independence hypothesis for the log-returns of Microsoft, using p = 6 consecutive observations and N = 1000 Monte Carlo samples.Also plot the dependogram.
Exercise 3.3(a) Which moment-based coefficients could we use to test the Gaussian distribution hypothesis? Are there other ways to verify the normality of a sample?(b) Are the following tests
Exercise 3.2 The MATLAB data set DataTentMap contains 1000 samples of size n =500 generated from the Tent Map series. Compute the percentage of rejection of the Ljung-Box test, using IndLBTest with
Exercise 3.1 Consider the tent map series defined by Xi+1 = 1 − |2Xi − 1| =2min(Xi, 1 − Xi), with X1 ∼ Unif(0, 1). Show that Xi ∼ Unif(0, 1) for all i ≥ 1. Also prove that the correlation
3. Construct a MATLAB program for the estimation of the value of a call-on-max option on the cumulative returns of two assets modeled by correlated Brownian motions, i.e., the payoff at maturity is
2. Construct a MATLAB program for the estimation of the value of an exchange option together with its greeks, using the generalization of the likelihood method of Broadie-Glasserman. The number of
1. Construct a MATLAB function for the estimation of the parameters of a Black-Scholes model with representation (2.4). The input values are a n × d matrix S representing the prices of the assets
Exercise 2.10 Suppose that a stock traded in France satisfies the stochastic differential equation dS(t) = 0.04S(t)dt + 0.15S(t)dW(t), while the EUR/CAD’s exchange rate process satisfies dC(t) =
Exercise 2.9 What is the relationship between the value C12 of an exchange option with payoff max(q1s1−q2s2, 0) and the value C21 of an exchange option with payoff max(q2s2 − q1s1, 0), when the
Exercise 2.8 Suppose that two stocks satisfy the Black-Scholes model with parametersμ1 = 0.04, μ2 = 0.09, σ1 = 0.25, σ2 = 0.35, and R12 = 0.25. We want to price an exchange option whose payoff at
Exercise 2.7 Compute the greeks (delta, gamma, vega) for an exchange option. How do they relate to the greeks of a call option on one asset? How much does one have to invest in the risk-free asset to
Exercise 2.6 Try to estimate the parameters of the bivariate Black-Scholes model for Apple and Microsoft, using the MATLAB function EstBS2DCor with the starting points μ1 = μ2 = 0 = ρ and σ1 =
Exercise 2.5 For a trivariate Black-Scholes model, show that if a is the Cholesky decomposition of the covariance matrix Σ, then aij has the same values as in the bivariate case for 1 ≤ j ≤ 2,
Exercise 2.4 Suppose that stock A and stock B follow the Black-Scholes model. With the associated daily returns, we computed the following volatility vector v =(0.011348, 0.005345, 0.013248).
Exercise 2.3 For a bivariate Black-Scholes model, show that the Cholesky decomposition a of the covariance matrix Σ is given byin terms of the volatilities σ1, σ2 and the correlation R = R12. Can
Exercise 2.2 Suppose that the covariance matrix Σ of 3 sets of returns is given byFind the Cholesky decomposition a of Σ, i.e., aa = Σ. 1.00 -0.68 0.12 = -0.68 0.58 -0.09 0.12 -0.09 0.23
Exercise 2.1 If W1, ...,Wd are correlated Brownian motions, show that Cov {Wj (s),Wk (t)} = Rjk min(s, t), s,t≥0, j,k∈ {1, . . . , d}.
3. Construct a MATLAB function for the computation of the implied volatility surface. The input must be a structure like the one of AppleCalls.The number of maturities must be deduced from the size
2. Use the MATLAB functions NumJacobian, NumHessian, and FormulaBS to compute the greeks of a call option. For a wide range of strikes, maturities, and stock prices, compare the numerical values with
1. Write a MATLAB function for estimating the greeks (delta, gamma, theta, rho, vega) of a European put option, using the two methods proposed by Broadie and Glasserman [1996]. Using N = 100000,
For a European put, write the MATLAB expressions for its value and the greeks (Δ, Γ, V), by using the likelihood ratio method of Broadie-Glasserman, as a function of the following variables:• s :
Let X1,X2, . . . , Xn be i.i.d. Gaussian with mean μ and variance σ2.(a) Show that the mean and variance of Y1 = |X1 − X2| are σ4π and 2σ2 1 − 2π respectively.(b) Prove that the
(a) Explain what is the implied volatility.(b) Find the implied volatility of a 9-month maturity European call option with a market value of $1.12, where the actual price of the asset is S0 = $73,
(a) Compute the greeks for a European binary call option.(b) Compute the greeks for a European put option.(c) For a European option with payoff Φ{S(T )}, show that Γ = V s2στ .Hint: Use
Suppose that S ∈ [$50, $150], K = $100, σ = 0.25, r = 1.5%, and τ = 1.(a) Plot the graph of Δ for the call and the put, as a function of the underlying asset.(b) Plot the graph of Γ for the
Let C be the value of a European call option given by the Black-Scholes formula.Prove that each of the following limits exist and give a financial interpretation of each result.(a) limσ→0 C(t,
Consider that the Black-Scholes model is valid for asset S, and assume it has a 36% volatility per annum. The risk-free rate is 2% and the actual value of the asset is $71.(a) We buy 100 3-month
Can you explain the difference between the results of EstBS1Dexp with the explicit method vs the results of EstBS1DNum with the numerical method?
Suppose that the daily price S of an asset satisfies the Black-Scholes model with parameters μ and σ (on a daily time scale). With the daily data, we first compute the log-returns xt, t = 1, . . .
State the main assumptions of the Black-Scholes model. Do you think they are met in practice?
This problem is a continuation of Problem 1 and therefore the information obtained from answering Problem 1 will also be used in this problem. Carry out the activities as required below.a. Enter the
Amortisation is specifically associated with:A. Debt instrumentsB. Equity instrumentsC. Discounted value of equity instrumentsD. Equity and debt instruments
An accountant has extracted the following information related to the company:The accountant’s estimate of ending retained earnings would be:A. €25 millionB. €95 millionC. €5 million
The financial accountant anticipates that the company will have assets of €25,000 at year-end and liabilities of €18,500. The accountant’s forecast of total owners’ equity should be about:A.
Which of the following would probably be classified as an investment activity?A. Issuance of debenture stockB. Acquisition of a competitorC. Selling of obsolete equipment

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